A birthday cake with candles
A birthday party was an event held in the honor of an individual, celebrating the anniversary of the day on which they were born.
Archive footage found by the crew of Enterprise NX-01 on Trialas IV showed what appeared to be the ninth birthday party of one of the Augment children living there. (ENT: "Cold Station 12")
Curzon Dax missed his own 100th birthday party, planned by Benjamin Sisko over the course of three months. (DS9: "Rejoined")
Commander Riker celebrated his birthday on the USS Enterprise-D with a birthday party. (TNG: "Future Imperfect")
In 2370, a replicant of Miles O'Brien was concerned the crew of Deep Space 9 were going to give him a surprise birthday party. (DS9: "Whispers")
Several months later, Riker wanted to give Worf a surprise party for his birthday, but Deanna Troi talked him out of it. (TNG: "Parallels")
For Kes' second birthday in 2371, the USS Voyager crew threw a surprise party for her. They explained the event (Kes found it exhilarating) and instructed her on the Earth tradition of making a wish and blowing out candles. Neelix made a seven layer Jimbalian fudge cake with l'maki nut icing and punch. Tom Paris replicated a locket as a gift. (VOY: "Twisted")
When Rom and the employees of Quark's gathered to strike in 2372, Quark mentioned that if they were having a party for him, his birthday was last month. (DS9: "Bar Association")
Keiko O'Brien prepared a surprise birthday party for her husband Miles in 2373, but during the party she actually was possessed by a pah-wraith. For the party she prepared q'parol, which was a big success. (DS9: "The Assignment")
In 2373, Ensign Ahni Jetal celebrated her birthday on the USS Voyager. (VOY: "Latent Image")
Neelix once recounted a happy memory while he and Tuvok were alone in a Jefferies tube, when Voyager's crew held his birthday party in the mess hall. (VOY: "The Haunting of Deck Twelve")
In an alternate timeline, the crew of Voyager held Kes' ninth birthday party in the ship's mess hall in 2378. (VOY: "Before and After")